Iwahori-Hecke algebras acting on tensor space by $q$-deformed letter permutations and $q$-partition algebras

Abstract

Let $R$ be a commutative ring with identity and let $V$ be a free $R$-module of rank $n$ for some $n\in \mathbb{N}$. Fixing an $R$-basis $\mathcal{E}$ of $V$, the symmetric group ${\mathfrak{S}}_n$ acts on $V$ by permuting $\mathcal{E}$ and hence on tensor space $V^{\otimes r}$ for $r\in \mathbb{N}$ via the usual tensor product action turning $V$ and $V^{\otimes r}$ into $R{\mathfrak{S}}_n$-modules. For units $q$ in $R$ we construct an action of the corresponding Iwahori-Hecke algebra ${\mathcal{H}}_{R,q}({\mathfrak{S}}_n)$ which specializes to the action of $R{\mathfrak{S}}_n$, if $q$ is taken to $1$. The centralizing algebra of this action is called the $q$-partition algebra ${\mathcal{P}}_{R,q}(n,r)$. Let $R$ be a field of characteristic not dividing $q$. We prove, that ${\mathcal{P}}_{R,q}(n,r)$ is isomorphic to the $q$-partition algebra defined by Halverson and Thiem by different means a few years ago.

21 pages, To appear in Journal of algebra